On degree bounds of $k$-uniform hypergraphs with bounded matching number

Abstract

We study the connection between the degree sequence of a $k$-uniform hypergraph and the size of its largest matching. Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices and let $d_1 \ge d_2 \ge \dots \ge d_n$ be the vertex degrees arranged in non-increasing order. For integers $k\ge 2$, $s\ge 2$ and $n > 2sk$, we prove that if the $(2sk+1)$-th largest degree satisfies $d_{2sk+1} > \binom{n-1}{k-1} - \binom{n-s}{k-1},$ then $\mathcal{F}$ contains a matching of size at least $s$. This can be viewed as a generalization of theorems by Lu, Guo, and Jiang (2023) and Huang and Rao (2026). Moreover, by relaxing the range of $n$, we obtain the same bound for the $(k+2s-2)$-th largest degree vertex. Note that the number $k+2s-2$ is optimal. For a $k$-set of vertices $S \subseteq [n]$, the degree of $S$ is defined as $\mathrm{deg}(S) = \sum_{v \in S} \mathrm{deg}(v)$, and the minimum of $\mathrm{deg}(S)$ over all non-edge $k$-subsets $S \notin E(\mathcal{F})$ of $V(\mathcal{F})$ is the Ore-degree of $\mathcal{F}$, denoted by $σ_k(\mathcal{F})$. Balogh, Palmer and Raeisi proved: for $s \ge 2$ and $n \ge 3k^2(s-1)$, if $σ_k(\mathcal{F}) > k\left(\binom{n-1}{k-1} - \binom{n-s}{k-1}\right),$ then $\mathcal{F}$ contains a matching of size $s$. They also conjectured that the result holds when $n > sk$. As a corollary, we prove that the bound on $n$ can be taken to be linear in $sk$ ($ n \geq 3sk $).